What is the derivative of ln(sqrt(sin(3^{x^2)})) ?

The derivative of ln(sqrt(sin(3^{x^2)})) is ln(3)*3^{x^2}xcot(3^{x^2})

What is the first derivative of ln(sqrt(sin(3^{x^2)})) ?

The first derivative of ln(sqrt(sin(3^{x^2)})) is ln(3)*3^{x^2}xcot(3^{x^2})

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\frac{d}{d x} (ln (sin (3^{x^{2}})))

ln (3) \cdot 3^{x^{2}} x cot (3^{x^{2}})

Solution steps

\frac{d}{d x} (ln (sin (3^{x^{2}})))

Apply the chain rule:

\frac{1}{sin ( 3 ^{x^{2}} )} \frac{d}{d x} (sin (3^{x^{2}}))

= \frac{1}{sin ( 3 ^{x^{2}} )} \frac{d}{d x} (sin (3^{x^{2}}))

\frac{d}{d x} (sin (3^{x^{2}})) = \frac{1}{2 sin ( 3 ^{x^{2}} )} cos (3^{x^{2}}) \cdot 2 ln (3) \cdot 3^{x^{2}} x

= \frac{1}{sin ( 3 ^{x^{2}} )} \cdot \frac{1}{2 sin ( 3 ^{x^{2}} )} cos (3^{x^{2}}) \cdot 2 ln (3) \cdot 3^{x^{2}} x

Simplify

\frac{1}{sin ( 3 ^{x^{2}} )} \cdot \frac{1}{2 sin ( 3 ^{x^{2}} )} cos (3^{x^{2}}) \cdot 2 ln (3) \cdot 3^{x^{2}} x : ln (3) \cdot 3^{x^{2}} x cot (3^{x^{2}})

= ln (3) \cdot 3^{x^{2}} x cot (3^{x^{2}})

(x^{2} + y^{2}) d x - 2 x y d y = 0

1 - ln (3 x)

f (x) = 3 x^{2}, at x = 1

\frac{d}{d x} (x (4 x + 1)^{\frac{1}{3}})

\int \frac{t}{t - 2} d t

What is the derivative of ln(sqrt(sin(3^{x^2)})) ?
The derivative of ln(sqrt(sin(3^{x^2)})) is ln(3)*3^{x^2}xcot(3^{x^2})
What is the first derivative of ln(sqrt(sin(3^{x^2)})) ?
The first derivative of ln(sqrt(sin(3^{x^2)})) is ln(3)*3^{x^2}xcot(3^{x^2})